The Virial Theorem is an important theorem in mechanics. There are various corollaries of the theorem which are sometimes also labeled the Virial Theorem. Those corollaries will be dealt with in due course. In the material below the variables which are vectors will be displayed in red.

Consider a system of n point particles indexed by i. Let r_i, v_i and p_i be the position, velocity and momentum vectors, respectively, for the i-th particle. Its mass is denoted as m_i. Let F_i be the net force, internal and external, impinging upon the i-th particle.

• Statement of the Virial Theorem: For the n point particles bound together into a system the time average of the kinetic energy of the particles, Σ½m_iv_i², plus one half of the time average of ΣF_i·r_i is equal to zero.

Proof

For each particle

v_i = dr_i/dt
p_i = m_iv_i
dp_i/dt = F_i

Define H as Σp_i·r_i. (Note that the dot product of two vectors is a scalar.) Then

dH/dt = Σ(dp_i/dt)·r_i + Σ p_i·(dr_i/dt)

Because (dp_i/dt) = F_i the first term on the right in the above equation reduces to ΣF_i·r_i.

Because p_i=m_iv_i =m_i(dr_i/dt) the second term on the right in the previous equation reduces to

Σ p_i·(dr_i/dt) = Σ m_i(dr_i/dt)²) =Σm_iv_i²

This last expression is just twice the kinetic energy K of the system; i.e., 2Σ½m_iv_i². Thus

dH/dt = ΣF_i·r_i + 2K

The Time Average of the Variables

The time average of a variable y(t) over the interval 0 to τ is defined as

y = (1/τ)∫₀^τ y(t)dt

Time averaging the equation of dH/dt gives

dH/dt = ΣF_i·r_i + 2K.

The time average of (dH/dt) is just

dH/dt = (1/τ)[H(τ)−H(0)]

If the system is cyclical such that it returns to its initial state after an interval then τ can be chosen equal to the cycle period and dH/dt reduces to 0. If the system is not cyclical then for the system being bounded the limit of dH/dt as τ increases without bound is zero.

Thus

ΣF_i·r_i + 2K = 0
or, equivalently
 
K + ½ΣF_i·r_i = 0

Corollaries

If the forces are generated as the gradients of a potential V(r₁,…,r_n) then

F_i = −∂V/∂r_i
and hence
K − ½(∂V/∂r_i)·r_i = 0

There is an important class of functions which are homogeneous. That is to say, if all of the arguments of the function are multiplied by a factor λ then the value of the function is multiplied by λ to some power, say m. The value m is said to be the degree of homogeneity of the function. For such functions the sum of the partial derivatives with respect to the arguments multiplied by the value of the arguments is just equal to m times the value of the function. This is called Euler's Theorem.

For forces which obey an inverse distance squared law the potential is just inversely proportional to the distance. Such potential functions are homogeneous to degree −1. For systems held together by mutual gravitation or electrostatic attraction the Virial Theorem reduces to

K + ½ V = 0
or, equivalently
K = − ½ V

For systems held together by mutual gravitational attraction the potential energy is negative so the kinetic energy is positive. The average total energy of the system T=K+V is given by

T = ½V

Electrons in Atoms

The relationship

K = − ½ V

also holds for the electrons in an atom, a system held together by the electrostatic force.

In atoms the electrons can change orbits going from a higher potential energy orbit to a lower one. When an electron does so the relationship that is relevant is

ΔK = − ½Δ V

When an electron loses potential energy only half of it goes into increased kinetic energy. The other half of the energy loss goes into the emission of a photon.

Dark Matter Within Galactic Clusters

Astronomy has made great use of the Virial Theorem as a way of measure gravitational mass. Consider a set of n galaxies each of mass m. Let v² be the measured time averaged squared velocity of a galaxy and v² the average of this quantity over the n galaxies. Then the time averaged kinetic energy of the system is n[½mv²].

The gravitational potential for two galaxies separated by a distance R is then −Gm²/R, where G is the gravitational constant. Let 1/R be the cluster average of the time average of (1/R). There are n(n-1)/2 pairs of galaxies so the time averaged potential of the system is then −[n(n-1)/2][Gm²/R]. Then, according to the Virial Theorem,

nmv²/2 = ½[n(n-1)/2](Gm²)(1/R)
which may be solved for m
giving
m = 2v²R/G(n-1)
and thus the total mass nm
of the cluster is
nm = [2v²/G]R[n/(n-1)]

where R is the reciprocal of (1/R).

In some clusters n is on the order of 1000 so n/(n-1) is essentially unity.

When this method has been applied to galactic clusters the gravitational mass computed is vastly greater than the measured mass derived from the amount of light generated by the stars in the galaxies. The difference is ascribed to dark matter.

Limitations

It is sometimes noted that virial theorems are powerful but dangerous theorem; dangerous in the sense that that they may easily be misapplied. Their misapplication then involves stating in untrue propositions as being true as a result of a virial theorem.

The conventional virial theorem applies only to point particles. If bodies are not point particles but have size and shape then some of the energy changes that take place when they interact goes into spin and perhaps distortions of the fundamental bodies.

Perhaps the most important limitation is that virial theorems do not apply to particles that are not in a bounded system. In particular it does not apply to the particles of an explosion. Thus applying a virial theorem to galactic clusters which are the unvirialized remnants of some primodial Big Bang could lead to absurdities.

(To be continued.)